Colloquium di Matematica - Pierre Cartier

"About a quantization of Galois Theory"

Pierre Cartier (Institut des Hautes Etudes Scientifiques, France)

Abstract

We begin by reviewing the Picard-Vessiot theory for differential equations , in the various presentations : monodromy , geometrical presentations using principal bundles and Lie groupoids , algebraic version using differential ring extensions , tensor categories . By analogy , we consider the equations using the Jackson q-derivations . Here also , there is a Galois theory , but for this purpose , we must replace ordinary groups by quantum groups as described by I. Manin . This is a continuation of recent work by Hiroshi Umemura (Nagoya University , Japan) . We use heavily Hopf algebraic methods to access to this noncommutative world.